1. 4.4 Proving Triangles are Congruent: ASA and AAS
Geometry

2. Objectives:
Prove that triangles are congruent using the ASA Congruence Postulate and the AAS Congruence Theorem
Use congruence postulates and theorems in real-life problems.

3. Assignment
Worksheet
Quiz after this section

4. Postulate 21: Angle-Side-Angle (ASA) Congruence Postulate
If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the triangles are congruent.

5. Theorem 4.5: Angle-Angle-Side (AAS) Congruence Theorem
If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of a second triangle, then the triangles are congruent.

6. Theorem 4.5: Angle-Angle-Side (AAS) Congruence Theorem
Given: A  D, C  F, BC  EF
Prove: ∆ABC  ∆DEF

7. Theorem 4.5: Angle-Angle-Side (AAS) Congruence Theorem
You are given that two angles of ∆ABC are congruent to two angles of ∆DEF. By the Third Angles Theorem, the third angles are also congruent. That is, B  E. Notice that BC is the side included between B and C, and EF is the side included between E and F. You can apply the ASA Congruence Postulate to conclude that ∆ABC  ∆DEF.

8. Ex. 1 Developing Proof
Is it possible to prove the triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.

9. Ex. 1 Developing Proof
A. In addition to the angles and segments that are marked, EGF JGH by the Vertical Angles Theorem. Two pairs of corresponding angles and one pair of corresponding sides are congruent. You can use the AAS Congruence Theorem to prove that ∆EFG  ∆JHG.

10. Ex. 1 Developing Proof
Is it possible to prove the triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.

11. Ex. 1 Developing Proof
B. In addition to the congruent segments that are marked, NP  NP. Two pairs of corresponding sides are congruent. This is not enough information to prove the triangles are congruent.

12. Ex. 1 Developing Proof
Is it possible to prove the triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.
UZ ║WX AND UW
║WX. 1 2 3 4

13. Ex. 1 Developing Proof
The two pairs of parallel sides can be used to show 1  3 and 2  4. Because the included side WZ is congruent to itself, ∆WUZ  ∆ZXW by the ASA Congruence Postulate. 1 2 3 4

14. Ex. 2 Proving Triangles are Congruent
Given: AD ║EC, BD  BC
Prove: ∆ABD  ∆EBC
Plan for proof: Notice that ABD and EBC are congruent. You are given that BD  BC
. Use the fact that AD ║EC to identify a pair of congruent angles.

15. Proof: Statements: BD  BC AD ║ EC D  C ABD  EBC ∆ABD  ∆EBC
Reasons: 1.

16. Proof: Statements: BD  BC AD ║ EC D  C ABD  EBC ∆ABD  ∆EBC
Reasons:
1. Given

17. Proof: Statements: BD  BC AD ║ EC D  C ABD  EBC ∆ABD  ∆EBC
Reasons:
Given

18. Proof: Statements: BD  BC AD ║ EC D  C ABD  EBC ∆ABD  ∆EBC
Reasons:
Given
Alternate Interior Angles

19. Proof: Statements: BD  BC AD ║ EC D  C ABD  EBC ∆ABD  ∆EBC
Reasons:
Given
Alternate Interior Angles
Vertical Angles Theorem

20. Proof: Statements: BD  BC AD ║ EC D  C ABD  EBC ∆ABD  ∆EBC
Reasons:
Given
Alternate Interior Angles
Vertical Angles Theorem
ASA Congruence Theorem

21. Note:
You can often use more than one method to prove a statement. In Example 2, you can use the parallel segments to show that D  C and A  E. Then you can use the AAS Congruence Theorem to prove that the triangles are congruent.